Mathematical modeling of gait on a movable horizontal base

Purpose of research the is development of a mathematical model describing human biomechanical and anthropometric parameters during walking under different conditions (slow and fast gait). The developed model is intended for integration into the control system of rehabilitation exoskeletons to enable dynamic correction of movement patterns through real-time processing of kinematic data. An additional objective is the parameterization of the model for adaptation to individual user characteristics.

Methods. Experimental studies involved video recording of key lower limb point movements of subjects on a treadmill, with subsequent data processing in OpenSourcePhysicsTracker software. A comparative analysis of trajectory approximation accuracy was conducted using polynomial regression and harmonic analysis (Fourier series) in the Curve Fitting Toolbox (MATLAB R2023a). Model validation was performed by calculating the mean square error (MSE), where the MSE did not exceed 1,51×10⁻² m², along with empirical methods.

Results. It was established that the trigonometric method (Fourier series) provides significantly higher accuracy in approximating periodic gait trajectories compared to the polynomial method, as confirmed by lower mean square error values. The polynomial model demonstrated unstable behavior at orders above 7, showing a tendency for significant deviations at the interval endpoints. For the harmonic model, the optimal number of components was 5–7 harmonics. Smooth approximated trajectories were obtained for all key points of the foot and the rotation angle of the metatarsophalangeal joint, with Fourier series expansion coefficients presented for coordinates along the X and Z axes.

Conclusion. An effective methodology for mathematical modeling of foot movement trajectories during walking based on Fourier series has been developed. This method is recognized as the most preferable for describing biomechanical walking patterns. The obtained models possess high application potential for creating control systems for rehabilitation equipment, enabling personalization based on patients' anthropometric characteristics.

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